Stampacchia Generalized Vector Quasiequilibrium Problems and Vector Saddle Points

In this paper, Stampacchia generalized vector quasiequilibrium problem and generalized vector loose saddle points for set-valued mappings are introduced. By using the scalarization method and the fixed-point theorem, existence theorems are established.This work was supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province, China. The author is grateful to Professor F. Giannessi and the referee for valuable comments and careful reading improving the original draft.Communicated by F. Giannessi     

Image Space Analysis and Scalarization of Multivalued Optimization

Using a new method based on generalized sections of feasible sets, we obtain optimality conditions for vector optimization of objective multifunctions with multivalued constraints. The authors express their sincere gratitude to Professor F. Giannessi and the referees for comments and valuable suggestions. The second author was partially supported by the Center of Excellence for Mathematics (University of Isfahan).     

Well Posedness in Vector Optimization Problems and Vector Variational Inequalities

In this paper, we give notions of well posedness for a vector optimization problem and a vector variational inequality of the differential type. First, the basic properties of well-posed vector optimization problems are studied and the case of C-quasiconvex problems is explored. Further, we investigate the links between the well posedness of a vector optimization problem and of a vector variational inequality. We show that, under the convexity of the objective function f, the two notions coincide. These results extend properties which are well known in scalar optimization. Communicated by F. Giannessi     

A convergence for vector valued functions

We give a definition of gamma-convergence (epi-convergence) for the case of vector valued sequences of functions and prove some related properties analogous to the scalar case. We obtain one of the main variational theorems about convergence of ?-minimizers. In the convex case, we consider also variable domains and obtain stability of efficient points and minimal values.     

Continuity of the Solution Map in Quadratic Programs under Linear Perturbations

It is well known that the solution map of a quadratic program where only the linear part of the data is subject to perturbation is an upper Lipschitz multifunction. This paper characterizes the continuity and lower semicontinuity of that solution map.This work was supported by the Brain Korea 21 Project in 2003, the APEC Postdoctoral Fellowships Program, and the KOSEF Brain Pool Program. The authors thank Professor F. Giannessi and two anonymous referees for helpful comments.Communicated by F. Giannessi     

On Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, we study the solvability for a class of vector variational inequalities in reflexive Banach spaces. By using Brouwer fixed point theorem, we prove the solvability for this class of vector variational inequalities without monotonicity assumption. The solvability results for this class of vector variational inequalities with monotone mappings are also presented by using the KKM-Fan lemma This paper is dedicated to Professor Franco Giannessi for his 68th birthday     

Upper semicontinuity of the solution maps in homogeneous vector quasi-equilibrium problems

The aim of this work was to study the solution maps in the vector homogeneous quasi-equilibrium problems. Here, we establish some sufficient and necessary conditions for upper semicontinuity of the solution maps in the vector homogeneous quasi-equilibrium problems. The results presented extend and improve the preceding results of Oetlli and Yen [Oetlli, W. and Yen, N.D., 1995, Continuity of the solution set of homogeneous equilibrium problems and linear complementarity problems. In: F. Giannessi and A. Maugeri (Eds) Variational Inequalities and Network Equilibrium Problems (New York: Plenum Press)].     

Feasibility and Solvability for Vector Complementarity Problems1

The purpose of this paper is to discuss the feasibility and solvability of vector complementarity problems. We prove that, under suitable conditions, the vector complementarity problem with a pseudomonotonicity assumption is solvable whenever it is strictly feasible. By strengthening the generalized monotonicity condition, we show also that the homogeneous vector complementarity problem is solvable whenever it is feasible. At last, we study the solvability of the vector complementarity problem on product spaces.The authors thank Professor Franco Giannessi for valuable comments and suggestions leading to improvements of this paper.     

Some remarks on the Minty vector variational principle

In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93-99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193-201]. In these papers, in the particular case of a differentiable objective function f taking values in Rm and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions.     

A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. In this paper we introduce a nonlinear scalarization function for a variable domination structure. Several important properties, such as subadditiveness and continuity, of this nonlinear scalarization function are established. This nonlinear scalarization function is applied to study the existence of solutions for generalized quasi-vector equilibrium problems. This paper is dedicated to Professor Franco Giannessi for his 68th birthday     

Sufficiency in Nonsmooth Multiobjective Programming Involving Generalized (Fρ)-convexity

In this paper, we consider a generalization of convexity for nonsmooth multiobjective programming problems. We obtain sufficient optimality conditions under generalized (Fρ)-convexity.This work was supported by Project 821134 and by the Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran.Communicated by F. Giannessi     

Duality Theorem for a Three-Phase Partition Problem

In some nonlinear diffusive phenomena, the systems have three or more stable states. Sternberg and Zeimer established the existence of minimal solutions for the problem of partitioning a certain domain Ω⊂ℝ² into three subdomains having least interfacial area. Ikota and Yanagida investigated stability and instability for stationary curves with one triple junction and for stationary binary-tree type interfaces. In this paper, we introduce a new concept of separation of three convex sets by a triangle, define a dual problem to the three-phase partition problem, and present a duality theorem. The author thanks Professor F. Giannessi for valuable comments, especially on Gale and Klee-type separation theorems. This research was partially supported by Kyushu University 21st Century COE Program (Development of Dynamic Mathematics with High Functionality) and by the Grant-in-Aid for General Scientific Research from the Japan Society for the Promotion of Science 14340037.     

A convergence for infinite dimensional vector valued functions

By using the definition of Γ-convergence for vector valued functions given in Oppezzi and Rossi (Optimization, to appear), we obtain a property of infimum values of the Γ-limit. By generalizing Mosco convergence to vector valued functions, we also obtain, in the convex case, the extension of some stability results analogous to the ones in Oppezzi and Rossi (optimization, to appear), when domain and value space are infinite dimensional.      

Infinite Time-Dependent Network Equilibria with a Multivalued Cost Function

In this paper, we consider the dynamic traffic network equilibria with possibly an infinite number of routes, a possibly multivalued cost function, and a not necessarily reflexive Banach space of flow trajectories. We investigate the existence of equilibria under a monotonicity assumption on the cost function, as well as an equivalent condition for equilibria under additional constraints. Finally, we give an iterative method for the computation of equilibria. Our results generalize and extend previous results in Refs. 1–2.Communicated by F. GiannessiThe authors are thankful to the referee and Professor F. Giannessi for valuable suggestions and comments leading the paper to its present form.All correspondence should be sent to the first author.     

Technical Note Discontinuous Implicit Quasivariational Inequalities in Normed Spaces

In this paper, we consider an implicit quasivariational inequality without continuity assumptions in normed spaces. The main result (Theorem 2.1) provides an infinite-dimensional version of Theorem 3.2 in Ref. 1. To achieve such a goal, we employ Theorem 3.2 in Ref. 1 and the technique of Cubiotti in Ref. 2. In particular, Theorem 3.1 covers a recent result of Cubiotti (Theorem 3.1 of Ref. 2) as a special case. Communicated by F. Giannessi This research was partially supported by the National Science Council of Taiwan, ROC.     

Semi-B-Preinvex Functions

In this note, a class of functions, called semi-B-preinvex function, which are a generalization of the semipreinvex functions and the B-vex functions, is introduced. Examples are given to show that there exist functions which are semi-B-preinvex functions, but are neither semipreinvex nor B-vex. A property of the semi-B-preinvex functions is obtained.Communicated by F. GiannessiThis research was partially supported by a Science Committee Project, Research Foundation of Chongqing, Grant 8409. The authors are thankful to the referees and Prof. F. Giannessi for suggestions and comments which helped to give the present form to this paper.     

Generalized Differential Properties of the Auslender Gap Function for Variational Inequalities

In this note, the Auslender gap function, which is used to formulate a variational inequality into an equivalent minimization problem, is shown to be differentiable in the generalized sense and has a lower contingent derivative under suitable conditions. This enables us to establish necessary and sufficient conditions for the existence of a solution to problems of variational inequalities.This research was partially supported by the National Natural Science Foundation of China and the Research Committee of Hong Kong Polytechnic University. Communicated by F. Giannessi     

Global Convergence of a Robust Smoothing SQP Method for Semi-Infinite Programming

The semi-infinite programming (SIP) problem is a program with infinitely many constraints. It can be reformulated as a nonsmooth nonlinear programming problem with finite constraints by using an integral function. Due to the nondifferentiability of the integral function, gradient-based algorithms cannot be used to solve this nonsmooth nonlinear programming problem. To overcome this difficulty, we present a robust smoothing sequential quadratic programming (SQP) algorithm for solving the nonsmooth nonlinear programming problem. At each iteration of the algorthm, we need to solve only a quadratic program that is always feasible and solvable. The global convergence of the algorithm is established under mild conditions. Numerical results are given. Communicated by F. Giannessi His work was supported by the Hong Kong Research Grant Council His work was supported by the Australian Research Council.     

Some Extensions of Nonconvex Economic Modeling: Invexity,Quasimax, and New Stability Conditions

There are many applications in economics of nonlinear programming, usually under convexity assumptions. Some nonconvex models have also been discussed extensively in order to relax the restrictive assumption of convexity. The applicability can be extended considerably by further generalization to invexity and of maximum to quasimax. Some qualitatively different effects may occur with nonconvex models, such as nonunique optima at different objective levels and jumps in the consumption function, which have economic significance. This paper describes these effects and relates them to mathematical concepts.Thanks are due to Professor F. Giannessi for correcting a number of details of thepresentation